Find the points on the number circle with the given abscissa. Coordinates. Property of point coordinates. Center of the number circle. From circle to trigonometer. Find the points on the number circle. Dots with abscissa. Trigonometer. Mark a point on the number circle. Number circle on the coordinate plane. Number circle. Points with ordinate. Give the coordinate of the point. Name the line and coordinate of the point.

““Derivatives” 10th grade algebra” - Application of derivatives to study functions. The derivative is zero. Find the points. Let's summarize the information. The nature of the monotonicity of the function. Application of the derivative to the study of functions. Theoretical warm-up. Complete the statements. Choose the correct statement. Theorem. Compare. The derivative is positive. Compare the formulations of the theorems. The function increases. Sufficient conditions for an extremum.

““Trigonometric equations” grade 10” - Values ​​from the interval. X= tan x. Provide roots. Is the equality true? Series of roots. Equation cot t = a. Definition. Cos 4x. Find the roots of the equation. Equation tg t = a. Sin x. Does the expression make sense? Sin x =1. Never do what you don't know. Continue the sentence. Let's take a sample of the roots. Solve the equation. Ctg x = 1. Trigonometric equations. The equation.

“Algebra “Derivatives”” - Tangent equation. Origin of terms. Solve a problem. Derivative. Material point. Differentiation formulas. Mechanical meaning of derivative. Evaluation criteria. Derivative function. Tangent to the graph of a function. Definition of derivative. Equation of a tangent to the graph of a function. Algorithm for finding the derivative. An example of finding the derivative. Structure of the topic study. The point moves in a straight line.

“Shortest path” - A path in a digraph. An example of two different graphs. Directed graphs. Examples of directed graphs. Reachability. The shortest path from vertex A to vertex D. Description of the algorithm. Advantages of a hierarchical list. Weighted graphs. Path in the graph. ProGraph program. Adjacent vertices and edges. Top degree. Adjacency matrix. Path length in a weighted graph. An example of an adjacency matrix. Finding the shortest path.

"The History of Trigonometry" - Jacob Bernoulli. Operating technique with trigonometric functions. The doctrine of measuring polyhedra. Leonard Euler. The development of trigonometry from the 16th century to the present day. The student has to meet trigonometry three times. Until now, trigonometry has been formed and developed. Construction common system trigonometric and related knowledge. Time passes, and trigonometry returns to schoolchildren.

The digits in multi-digit numbers are divided from right to left into groups of three digits each. These groups are called classes. In each class, the numbers from right to left indicate the units, tens and hundreds of that class:

The first class on the right is called class of units, second - thousand, third - millions, fourth - billions, fifth - trillion, sixth - quadrillion, seventh - quintillions, eighth - sextillion.

For ease of reading the recording multi-digit number, a small gap is left between classes. For example, to read the number 148951784296, we highlight the classes in it:

and read the number of units of each class from left to right:

148 billion 951 million 784 thousand 296.

When reading a class of units, the word units is usually not added at the end.

Each digit in the notation of a multi-digit number occupies a certain place - position. The place (position) in the record of a number on which the digit stands is called discharge.

The counting of digits goes from right to left. That is, the first digit on the right in a number is called the first digit, the second digit on the right is the second digit, etc. For example, in the first class of the number 148,951,784,296, digit 6 is the first digit, 9 is the second digit, 2 - third digit:

Units, tens, hundreds, thousands, etc. are also called bit units:
units are called units of the 1st category (or simple units)
tens are called units of the 2nd digit
hundreds are called 3rd digit units, etc.

All units except simple units are called constituent units. So, ten, hundred, thousand, etc. are composite units. Every 10 units of any rank constitutes one unit of the next (higher) rank. For example, a hundred contains 10 tens, a ten contains 10 prime ones.

Any compound unit compared to another unit smaller than it is called unit of the highest category, and in comparison with a unit greater than it is called unit of the lowest category. For example, a hundred is a higher-order unit relative to ten and a lower-order unit relative to a thousand.

To find out how many units of any digit there are in a number, you need to discard all the digits representing the units of lower digits and read the number expressed by the remaining digits.

For example, you need to find out how many hundreds there are in the number 6284, i.e. how many hundreds are in the thousands and hundreds of a given number together.

In the number 6284, the number 2 is in third place in the units class, which means there are two prime hundreds in the number. The next number to the left is 6, meaning thousands. Since every thousand contains 10 hundreds, 6 thousand contain 60 of them. In total, therefore, this number contains 62 hundreds.

The number 0 in any digit means the absence of units in this digit. For example, the number 0 in the tens place means the absence of tens, in the hundreds place - the absence of hundreds, etc. In the place where there is a 0, nothing is said when reading the number:

172 526 - one hundred seventy two thousand five hundred twenty six.
102 026 - one hundred two thousand twenty six.

These are the numbers that are used when counting: 1, 2, 3... etc.

Zero is not natural.

Natural numbers are usually denoted by the symbol N.

Whole numbers. Positive and negative numbers

Two numbers that differ from each other only by sign are called opposite, for example, +1 and -1, +5 and -5. The "+" sign is usually not written, but it is assumed that there is a "+" in front of the number. Such numbers are called positive. Numbers preceded by a "-" sign are called negative.

The natural numbers, their opposites and zero are called integers. The set of integers is denoted by the symbol Z.

Rational numbers

These are finite fractions and infinite periodic fractions. For example,

The set of rational numbers is denoted Q. All integers are rational.

Irrational numbers

An infinite non-periodic fraction is called an irrational number. For example:

The set of irrational numbers is denoted J.

Real numbers

The set of all rational and all irrational numbers is called set of real (real) numbers.

Real numbers are represented by the symbol R.

Rounding numbers

Consider the number 8,759123... . Rounding to the nearest whole number means writing down only the part of the number that is before the decimal point. Rounding to tenths means writing down the whole part and one digit after the decimal point; round to the nearest hundredth - two digits after the decimal point; up to thousandths - three digits, etc.

The concept of a real number: real number- (real number), any non-negative or negative number or zero. Real numbers are used to express measurements of each physical quantity.

Real, or real number arose from the need to measure geometric and physical quantities peace. In addition, for performing root extraction operations, calculating logarithms, solving algebraic equations, etc.

Natural numbers were formed with the development of counting, and rational numbers with the need to manage parts of a whole, then real numbers (real) are used to measure continuous quantities. Thus, the expansion of the stock of numbers that are considered led to the set of real numbers, which, in addition to rational numbers, consists of other elements called irrational numbers.

Set of real numbers(denoted R) are sets of rational and irrational numbers collected together.

Real numbers divided byrational And irrational.

The set of real numbers is denoted and often called real or number line. Real numbers consist of simple objects: whole And rational numbers.

A number that can be written as a ratio, wherem is an integer, and n- natural number, isrational number.

Any rational number can easily be represented as a finite fraction or an infinite periodic decimal fraction.

Example,

Infinite decimal, is a decimal fraction that has an infinite number of digits after the decimal point.

Numbers that cannot be represented in the form are ir rational numbers .

Example:

Any irrational number can easily be represented as an infinite non-periodic decimal fraction.

Example,

Rational and irrational numbers create set of real numbers. All real numbers correspond to one point on the coordinate line, which is called number line.

For numerical sets the following notation is used:

  • N- set of natural numbers;
  • Z- set of integers;
  • Q- set of rational numbers;
  • R- set of real numbers.

Theory of infinite decimal fractions.

A real number is defined as infinite decimal, i.e.:

±a 0 ,a 1 a 2 …a n …

where ± is one of the symbols + or −, a number sign,

a 0 is a positive integer,

a 1 ,a 2 ,…a n ,… is a sequence of decimal places, i.e. elements of a numerical set {0,1,…9}.

An infinite decimal fraction can be explained as a number that lies between rational points on the number line like:

±a 0 ,a 1 a 2 …a n And ±(a 0 ,a 1 a 2 …a n +10 −n) for all n=0,1,2,…

Comparison of real numbers as infinite decimal fractions occurs place-wise. For example, suppose we are given 2 positive numbers:

α =+a 0 ,a 1 a 2 …a n …

β =+b 0 ,b 1 b 2 …b n …

If a 0 0, That α<β ; If a 0 >b 0 That α>β . When a 0 =b 0 Let's move on to the comparison of the next category. Etc. When α≠β , which means that after a finite number of steps the first digit will be encountered n, such that a n ≠b n. If a n n, That α<β ; If a n >b n That α>β .

But it is tedious to pay attention to the fact that the number a 0 ,a 1 a 2 …a n (9)=a 0 ,a 1 a 2 …a n +10 −n . Therefore, if the record of one of the numbers being compared, starting from a certain digit, is a periodic decimal fraction with 9 in the period, then it must be replaced with an equivalent record with a zero in the period.

Arithmetic operations with infinite numbers decimals it is a continuous continuation of the corresponding operations with rational numbers. For example, the sum of real numbers α And β is a real number α+β , which satisfies the following conditions:

∀ a′,a′′,b′,b′′∈ Q(a′⩽ α ⩽ a′′)∧ (b′⩽ β ⩽ b′′)⇒ (a′+b′⩽ α + β ⩽ a′′+b′′)

The operation of multiplying infinite decimal fractions is defined similarly.


What is a number? NUMBER is one of the basic concepts of mathematics; it originated in ancient times and gradually expanded and generalized. In connection with the counting of individual objects, the concept of positive integer (natural) numbers arose, and then the idea of ​​​​the limitlessness of the natural series of numbers: 1, 2, 3. Natural numbers are numbers used in counting objects. 1


Story. During excavations of a camp of ancient people, a wolf bone was found, on which 30 thousand years ago, some ancient hunter made fifty-five notches. It is clear that while making these notches, he was counting on his fingers. The pattern on the bone consisted of eleven groups, each with five notches. At the same time, he separated the first five groups from the rest with a long line. Also in Siberia and other places, stone tools and decorations made in the same distant era were found, which also had lines and dots, grouped in 3, 5 or 7. Celts - ancient people, who lived in Europe 2500 years ago, who are the ancestors of the French and English, were considered twenties (two arms and two legs gave twenty fingers). Traces of this have been preserved in French, where the word “eighty” sounds like “four times twenty.” Other peoples also considered twenty - the ancestors of the Danes and Dutch, Ossetians and Georgians. 2




Even and odd numbers. An even number is an integer that is divisible by 2 without a remainder: ..., 2, 4, 6, 8, ... An odd number is an integer that is not divisible by 2 without a remainder: ..., 1, 3, 5, 7, 9, ... Pythagoras defining number as energy and believed that through the science of numbers the secret of the Universe is revealed, for number contains the secret of things. Even numbers Pythagoras considered the numbers to be female and the odd numbers to be male: 2+3=5 5 is a symbol of family, marriage. Even and odd numbers = female and male numbers. 4


Simple and compound. A prime number is a natural number that has exactly two distinct natural factors: one and itself. The sequence of prime numbers starts like this: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, ... Composite numbers are numbers that have 3 or more divisors. Number theory studies the properties of prime numbers. So everything integers More than one is divided into simple and compound. 5


Perfect and imperfect numbers. Perfect numbers, positive integers, equal to the amount all its regular (i.e., less than this number) divisors. For example, the numbers 6 = and 28 = are perfect. Until now (1976) not a single odd Owl is known. hours and the question of their existence remains open. Research about Sov. hours were started by the Pythagoreans, who attributed a special mystical meaning to numbers and their combinations. Pythagoras called imperfect numbers the sum of regular divisors that are less than himself. 6




Magic numbers. The secrets of numbers attract people, force them to delve into, understand, and compare their conclusions with the real relationship of affairs. To the numbers in ancient world They were very respectful. People who knew them were considered great, they were equated with deities. The simplest example is the absence in many countries of aircraft with tail number 13, floors and hotel rooms with number “13”. 8
Magic series 2 is the number of balance and contrast, and supports stability, mixing positive and negative qualities. 6 – Symbol of reliability. It is a perfect number that is divisible by both an even number (2) and an odd number (3), thus combining the elements of each. 8 – Number of material success. It means reliability brought to perfection, as it is represented by a double square. Divided in half, it has equal parts (4 and 4). If it is further divided, then the parts will also be equal (2, 2, 2, 2), showing a fourfold equilibrium. 9 – The number of universal success, the largest of all numbers. Like three times the number 3, nine turns instability into aspiration. 10